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Skin friction and pressure: the “footprints” of turbulence. (English) Zbl 1060.76068

Summary: The problems of exact state reconstruction and approximate state estimation based on wall information in a wall-bounded incompressible unsteady flow are addressed. It is shown that, if in an arbitrarily small neighborhood of time \(t\) precise measurements are made of the two components of wall skin friction and the wall pressure, all terms in the Taylor-series expansions of the unsteady flow state near the wall at time \(t\) may be determined (in the linear setting, this determination may be made based on skin-friction measurements alone). Combining this fact with the analyticity of solutions of the nonlinear Navier-Stokes equation and the unique continuation theorem for analytic functions, in theory complete reconstruction of a fully-developed turbulent flow in a channel at any Reynolds number at time \(t\) is possible given only information about the unsteady flow available at the wall in a neighborhood of time \(t\), without knowledge of the initial conditions of the flow.
In light of limitations on direct extrapolations from measurements in the practical setting, an adjoint-based algorithm is presented and numerically tested for estimating the state of an entire turbulent channel-flow system based on a time history of noisy measurements at the wall. This algorithm effectively uses the unsteady nonlinear Navier-Stokes equation itself as a filter to find the flow solution that is most consistent with the available measurements.

MSC:

76F70 Control of turbulent flows
35Q30 Navier-Stokes equations
35R30 Inverse problems for PDEs
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
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